Research indicates that a large percentage of students have misconceptions of the equal sign and that this can be detrimental to their mathematical pursuits as they progress through algebra. In order for students to develop a strong understanding of the meaning of the equal sign, algebra teachers must be aware of the viewpoints their students will hold pertaining to the equal sign and strategies to encourage a relational view of the equal sign and a progression of sophistication within the relational spectrum. In an algebra course for preservice teachers, we created a project where they investigated students’ conceptions of the equal sign and determined interventions to foster more productive views. The purpose of our study was to examine our preservice teachers’ ability to differentiate between the various students’ conceptions of the equal sign, to determine which teacher interventions resonated well with the preservice teachers, and to evaluate the overall effectiveness of the project. This article will also outline ideas for other mathematics teacher educators that wish to implement similar practice-based teaching experiences in their content courses and potential paths of future research.

A lesson study cycle is a professional development process that integrates research and reflection through collaboration. The cycle allows a group to refine a lesson based on these collaboration efforts such as interaction with students and the post-lesson discussion. Secondary pre-service teachers in a mathematics methods course engaged in a lesson study cycle through collaboration between in-service teachers, Georgia College professors, and students in a local high school classroom. We systematically investigated this process to determine that through preparing, enacting and reflecting on their practice, Pre-service Teachers (PST) developed insight, reasoning, and understanding of the mathematics that they taught.

Let G be the complex connected simply connected simple Lie group of type G2 or F4. Let K denote the fixed point subgroup relative to an involution of G that is lifted from a Cartan involution. This article gives a description of certain components of Springer fibers associated to closed K-orbits contained in the flag variety of G. These components allow us to describe certain multiplicity polynomials associated to discrete series representations of the real form G22 of G2 and the two real forms F44 and F4−20 of F4. The goals for this paper are motivated by the descriptions of Springer fiber components and the associated multiplicity polynomials for type SU(p,q) described in a paper of Barchini and Zierau.

Letkkbe an algebraically closed field of characteristicp>0p > 0, and letGGbe a simple, simply connected algebraic group defined overFp\mathbb {F}_p. Givenr≥1r \geq 1, setq=prq=p^r, and letG(Fq)G(\mathbb {F}_q)be the corresponding finite Chevalley group. In this paper we investigate the structure of the first cohomology groupH1⁡(G(Fq),L(λ))\operatorname {H}^1(G(\mathbb {F}_q),L(\lambda )), whereL(λ)L(\lambda )is the simpleGG-module of highest weightλ\lambda. Under certain very mild conditions onppandqq, we are able to completely describe the first cohomology group whenλ\lambdais less than or equal to a fundamental dominant weight. In particular, in the cases we consider, we show that the first cohomology group has dimension at most one. Our calculations significantly extend, and provide new proofs for, earlier results of Cline, Parshall, Scott, and Jones, who considered the special case whenλ\lambdais a minimal non-zero dominant weight.

We consider a generalization of the Frobenius problem, where the object of interest is the greatest integer having exactly j representations by a collection of positive relatively prime integers. We prove an analogue of a theorem of Brauer and Shockley and show how it can be used for computation.

A lesson study cycle is a professional development process that integrates research and reflection through collaboration. The cycle allows a group to refine a lesson based on these collaboration efforts such as interaction with students and the post-lesson discussion. Secondary pre-service teachers in a mathematics methods course engaged in a lesson study cycle through collaboration between in-service teachers, Georgia College professors, and students in a local high school classroom. We systematically investigated this process to determine that through preparing, enacting and reflecting on their practice, Pre-service Teachers (PST) developed insight, reasoning, and understanding of the mathematics that they taught. GAMTE Proceedings 2015 2 ENGAGING IN LESSON STUDY AT GEORGIA COLLEGE Students in the Methods of Secondary Mathematics Teachers course at Georgia College participated in a lesson study project. In this course, students examined research on instructional strategies, assessment techniques, lesson planning, multicultural and gender issues, beliefs, and student learning in mathematics. These pre-service teachers (PST) worked in groups in order to develop a lesson based on research. Through collaboration with an in-service secondary mathematics teacher (IST) at Georgia College’s Early College (GCEC), PST observed his class, created a lesson plan, taught the lesson, rewrote the lesson plan, and retaught the lesson. In this article, we investigate how engaging secondary PST in a lesson study will inform their perceptions of research, practice and students’ mathematics.